a. A rectangular pen is built with one side against a barn. 1900 m of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 msquared. What are the dimensions of each pen that minimize the amount of fence that must be used?

475 m, 950 m

Explanation:

Let l be the length of the side perpendicular to the barn.

1900-2l = length of the side parallel to the barn

Area A = l (1900-2l)

A = 1900l-2l^2

now, the maximum value of l (the equation being quadratic)

l_max = - b/2a

a = 2

b=1900

l_max = - 1900/4 = 475 m

then 1900-2l = 1900-2 * (475) = 950 m

So, the dimensions that maximize area are

950 and 475

Now. A_max = - 2 (l_max) ^2+1900*l_max

A_max = - 2 (475) ^2+1900*475

A_max = 451250 m^2

or, 475*950 = 451250 m^2